I doubt it is the rub rail, as it is horizontal - but it could be. You can pull the rubber out of the plastic keeper if you need to look at the screws. I'd guess it is coming through a deck fitting like say a stanchion, and flowing downhill along the ceiling and then dripping.
If I were you, I'd remove the 3 bolts holding the chain plate to the bulkhead and check it. You can see water stains, which leads me to believe the cover plate and slot on the deck are not fully sealed. The bulkhead is plywood, so if water gets in from above it'll soak into the end grain and split the plys. That happened to me, and the bulkhead cracked under sail - I was lucky in that the bolts prevented the chain plate from sliding up, and I lowered the sail immediately.
It is easy (your mast is down, right?) to remove the 4 screws on the cover plates (I'd check both sides), look at the sealing material or caulk, and unbolt the chain plate. If the bulkhead wood seems soft or swelled, I would replace it. I used exterior grade ply, cut it to the same shape, and sealed all edges with West Systems epoxy. You have to get the decorative plugs out of the fiddle on the shelf and unscrew it to expose the bulkhead, and peel off the grey fabric around there. You might have to sacrifice the plastic trim covering the edge of the fabric along the side of the bulkhead. Then it is a messy job to use a grinding wheel and/or an oscillating tool blade to cut out the bad bulkhead(s), being careful not to grind into the hull layup. Then I used thickened epoxy to "glue" the new bulkhead in place, with the chain plate metal loosely bolted to the wood and up through the slot to ensure the right positioning. After that kicked I filleted the edge between the wood and hull with more thickened epoxy, making a curved fillet. Then you need to tab the bulkhead back to the hull with glass cloth. I felt that the type with the bias at 45 degrees would be stronger than cloth where the strands align up and sown ad horizontally. It may not look so nice after (unless maybe you find teak veneer to reapply), but if you do it right you will have a working mast.